The question now is, has Kryptonite ever been discussed in this continuity? A search says that last post was the first time the word was ever used (don't really trust SHH searches though). If it hasn't been seen before, it's unlikely the government would know anything about, much less know it's name or effects on Kryptonians.
Of course, even if no one knew what Kryptonite was, Ollie could still have the arrow. He'd probably use it after none of his others arrows work, and everyone finds out about Kryptonite the hard way.
